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At least 1,261 records · Page 70

The influence of hydrothermal synthesis temperature on the electronic structure and luminescence property of Cr-doped ZnGa 2 O 4 nanoparticles

Cr-doped ZnGa 2 O 4 (CZGO) is a near-infrared-emitting material with long-lasting persistent luminescence. This unique property makes it a great candidate in optical imaging and sensing for various biomedical applications. Nanosized CZGO can be synthesized via a hydrothermal approach, but the reaction temperatures reported in existing literature vary from 120°C to 220°C, approaching the highest temperature allowed for a standard autoclave vessel. The reason for choosing a particular hydrothermal synthesis temperature has rarely been discussed, and it is unclear whether the resulting CZGO possess the same optical properties. This work compares CZGO nanoparticles synthesized at temperatures from the lowest reported 120°C and to as high as 220°C. We find that although all the synthesized CZGO are light emitting, they exhibit different luminescence intensities and respond differently to the change of excitation energy. X-ray absorption fine structure (XAFS) analysis is employed to investigate the local chemical environment around Zn, Ga and Cr, respectively. We found synthesis temperature strongly influences the Ga species formed in these particles. The energy transfer mechanism is further elucidated using X-ray excited optical luminescence (XEOL) in combination with element-specific XAFS. Two energy transfer paths are identified, which explains the different excitation energy dependencies of the observed photoluminescence.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Principles and applications of x-ray light sources driven by laser wakefield acceleration

One of the most prominent applications of modern particle accelerators is the generation of radiation. In a synchrotron or an x-ray free electron laser (XFEL), high energy electrons oscillating in periodic magnetic structures emit bright x rays. In spite of their scientific appeal that will remain evident for many decades, one limitation of synchrotrons and XFELs is their typical mile-long size and their cost, which often limits access to the broader scientific community. Here, this tutorial reviews the principles and prospects of using plasmas produced by intense lasers as particle accelerators and x-ray light sources, as well as some of the applications they enable. A plasma is an ionized medium that can sustain electrical fields many orders of magnitude higher than that in conventional radio frequency accelerator structures and can be used to accelerate electrons. When short, intense laser pulses are focused into a gas, it produces electron plasma waves in which electrons can be trapped and accelerated to GeV energies. This process, laser-wakefield acceleration (LWFA), is analogous to a surfer being propelled by an ocean wave. Many radiation sources, from THz to gamma-rays, can be produced by these relativistic electrons. This tutorial reviews several LWFA-driven sources in the keV-MeV photon energy range: betatron radiation, inverse Compton scattering, bremsstrahlung radiation, and undulator/XFEL radiation. X rays from laser plasma accelerators have many emerging applications. They can be used in innovative and flexible x-ray imaging and x-ray absorption spectroscopy configurations, for use in biology, industry, and high-energy density science.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Imaging plasma formation in isolated nanoparticles with ultrafast resonant scattering

We have recorded the diffraction patterns from individual xenon clusters irradiated with intense extreme ultraviolet pulses to investigate the influence of light-induced electronic changes on the scattering response. The clusters were irradiated with short wavelength pulses in the wavelength regime of different 4d inner-shell resonances of neutral and ionic xenon, resulting in distinctly different optical properties from areas in the clusters with lower or higher charge states. The data show the emergence of a transient structure with a spatial extension of tens of nanometers within the otherwise homogeneous sample. Simulations indicate that ionization and nanoplasma formation result in a light-induced outer shell in the cluster with a strongly altered refractive index. The presented resonant scattering approach enables imaging of ultrafast electron dynamics on their natural timescale.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Simulating environmentally-sensitive tree recruitment in vegetation demographic models

Vegetation demographic models (VDMs) endeavor to predict how global forests will respond to climate change. This requires simulating which trees, if any, are able to recruit under changing environmental conditions. We present a new recruitment scheme for VDMs in which functional-type-specific recruitment rates are sensitive to light, soil moisture and the productivity of reproductive trees. We evaluate the scheme by predicting tree recruitment for four tropical tree functional types under varying meteorology and canopy structure at Barro Colorado Island, Panama. We compare predictions to those of a current VDM, quantitative observations and ecological expectations. We find that the scheme improves the magnitude and rank order of recruitment rates among functional types and captures recruitment limitations in response to variable understory light, soil moisture and precipitation regimes. Furthermore, our results indicate that adopting this framework will improve VDM capacity to predict functional-type-specific tree recruitment in response to climate change, thereby improving predictions of future forest distribution, composition and function.

59 BASIC BIOLOGICAL SCIENCES↗

Controllable Phycobilin Modification: An Alternative Photoacclimation Response in Cryptophyte Algae

Cryptophyte algae are well-known for their ability to survive under low light conditions using their auxiliary light harvesting antennas, phycobiliproteins. Mainly acting to absorb light where chlorophyll cannot (500-650 nm), phycobiliproteins also play an instrumental role in helping cryptophyte algae respond to changes in light intensity through the process of photoacclimation. Until recently, photoacclimation in cryptophyte algae was only observed as a change in the cellular concentration of phycobiliproteins; however, an additional photoacclimation response was recently discovered that causes shifts in the phycobiliprotein absorbance peaks following growth under red, blue, or green light. Here, we reproduce this newly identified photoacclimation response in two species of cryptophyte algae and elucidate the origin of the response on the protein level. We compare isolated native and photoacclimated phycobiliproteins for these two species using spectroscopy and mass spectrometry, and we report the X-ray structures of each phycobiliprotein and the corresponding photoacclimated complex. We find that neither the protein sequences nor the protein structures are modified by photoacclimation. We conclude that cryptophyte algae change one chromophore in the phycobiliprotein β subunits in response to changes in the spectral quality of light. Ultrafast pump-probe spectroscopy shows that the energy transfer is weakly affected by photoacclimation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Structural color from a coupled nanowire pair beyond the bonding and antibonding model

Optical resonances in nanostructures can be harnessed to produce a wide range of structural colors. Conversely, the analysis of structural colors has been used to clarify the nature of optical resonances. Here, we show that silicon nanowire (NW) pairs can display a wide range of structural colors by controlling their radiative coupling. This is accomplished by exciting a series of Fabry–Pérot-like modes where light is repeatedly scattered between two NWs. These modes are beyond the expectation from the conventional chemical bonding model under a quasi-electrostatic approximation, in which only bonding and antibonding modes can be formed in a pair system through modal hybridization. The additional eigenmodes found in a two-resonator system originate from the nonlinear, frequency-dependent coupling strength derived from the radiative nature of low- Q resonators. The Fabry–Pérot modes can be tuned across the entire visible frequency range by varying the distance between two NWs, leading to what we believe is a new type of universal building blocks that can provide structural color within a subwavelength footprint. The presented results pave the way toward the design and usage of highly tunable resonances that exploit the radiative coupling of high-index nanostructures.

Li, Qitong↗

Methods for tuning plasmonic and photonic optical resonances in high surface area porous electrodes

Abstract Surface plasmons have found a wide range of applications in plasmonic and nanophotonic devices. The combination of plasmonics with three-dimensional photonic crystals has enormous potential for the efficient localization of light in high surface area photoelectrodes. However, the metals traditionally used for plasmonics are difficult to form into three-dimensional periodic structures and have limited optical penetration depth at operational frequencies, which limits their use in nanofabricated photonic crystal devices. The recent decade has seen an expansion of the plasmonic material portfolio into conducting ceramics, driven by their potential for improved stability, and their conformal growth via atomic layer deposition has been established. In this work, we have created three-dimensional photonic crystals with an ultrathin plasmonic titanium nitride coating that preserves photonic activity. Plasmonic titanium nitride enhances optical fields within the photonic electrode while maintaining sufficient light penetration. Additionally, we show that post-growth annealing can tune the plasmonic resonance of titanium nitride to overlap with the photonic resonance, potentially enabling coupled-phenomena applications for these three-dimensional nanophotonic systems. Through characterization of the tuning knobs of bead size, deposition temperature and cycle count, and annealing conditions, we can create an electrically- and plasmonically-active photonic crystal as-desired for a particular application of choice.

36 MATERIALS SCIENCE↗

Pentaquarks made of light quarks and their admixture to baryons

This paper is a continuation of our studies of multiquark hadrons. The antisymmetrization of their wave functions required by Fermi statistics is nontrivial, as it mixes orbital, color, spin, and flavor structures. In our previous papers we developed a method to find them based on the representations of the permutation group, and derived the explicit wave functions for baryons excited to the first and second shells (L = 1, 2), tetraquarks $qq$$\overline{q}$$\overline{q}$ and hexaquarks (6q). Now we apply it to light pentaquarks ($qqq$$\overline{q}$), in the S- and P-shells (L = 0, 1). Using Jacobi coordinates, one can use the hyperdistance approximation in 12-dimensional space. We further address the issue of “unquenching” of baryons, by considering their mixing with pentaquarks, via two channels, through the addition of σ-like or π-like $\overline{q}$$q$ pairs. This mixing is central for understanding of the observed flavor asymmetry of the antiquark sea, the amount of orbital motion issue as well as other nucleon properties.

Baryons↗

Photophysics of nanographenes: from polycyclic aromatic hydrocarbons to graphene nanoribbons

Graphene quantum dots (GQDs) and nanoribbons (GNRs) are classes of nanographene molecules that exhibit highly tunable photophysical properties. There have been great strides in recent years to advance our understanding of nanographene photophysics and develop their use in light-harvesting systems, such as artificial photosynthesis. Here, we review the latest studies of GQDs and GNRs which have shed new light onto their photophysical underpinnings through computational and advanced spectroscopic techniques. We discuss how the size, symmetry, and shape of nanographenes influence their molecular orbital structures and, consequentially, their spectroscopic signatures. The scope of this review is to comprehensively lay out the general photophysics of nanographenes starting with benzene and building up to larger polycyclic aromatic hydrocarbons, GQDs, and GNRs. We also explore a collection of publications from recent years that build upon the current understanding of nanographene photophysics and their potential application in light-driven processes from display, lasing, and sensing technology to photocatalytic water splitting.

59 BASIC BIOLOGICAL SCIENCES↗

Human CD4-binding site antibody elicited by polyvalent DNA prime-protein boost vaccine neutralizes cross-clade tier-2-HIV strains

The vaccine elicitation of HIV tier-2-neutralization antibodies has been a challenge. Here, we report the isolation and characterization of a CD4-binding site (CD4bs) specific monoclonal antibody, HmAb64, from a human volunteer immunized with a polyvalent DNA prime-protein boost HIV vaccine. HmAb64 is derived from heavy chain variable germline gene IGHV1-18 and light chain germline gene IGKV1-39. It has a third heavy chain complementarity-determining region (CDR H3) of 15 amino acids. On a cross-clade panel of 208 HIV-1 pseudo-virus strains, HmAb64 neutralized 20 (10%), including tier-2 strains from clades B, BC, C, and G. The cryo-EM structure of the antigen-binding fragment of HmAb64 in complex with a CNE40 SOSIP trimer revealed details of its recognition; HmAb64 uses both heavy and light CDR3s to recognize the CD4-binding loop, a critical component of the CD4bs. This study demonstrates that a gp120-based vaccine can elicit antibodies capable of tier 2-HIV neutralization.

60 APPLIED LIFE SCIENCES↗

Prototype Modeling for a Light-Trapping Planar-Cavity Enclosed Particle Solar Receiver

Concentrating solar thermal (CST) systems present a promising avenue for affordable and reliable energy production. Solar receivers are key components that determine the efficiency and longevity of these systems. Particle-based solar receivers have emerged as a compelling alternative to traditional technologies, offering several advantages that address limitations in current CST systems. This is especially true as next-generation CST technologies target applications including electricity generation, thermochemical processes, and industrial process heat, many of which necessitate higher operating temperatures than current commercial molten salt systems. Molten-salt thermal energy storage (TES) systems, commonly used in CSP, face challenges related to freezing and corrosion. Particle-based TES systems, in contrast, do not experience these issues, as particles are stable at high temperatures, exceeding 1000 degrees Celsius. This capability allows for a wider range of applications, including those requiring higher temperatures for industrial processes and efficient electricity generation. A novel innovation in particle-based solar receiver technology is the light-trapping planar cavity receiver (LTPCR) configuration developed by NREL. The LTPCR design consists of small cavity-like structures using opaque planar surfaces, enabling efficient capture and absorption of solar energy. A high incident flux concentration at the cavity aperture is absorbed on the receiver walls, and subsequently transferred to particles on the inside of cavities. The particles flow through the system, forming a fluidized bed inside of the receiver panels, effectively capturing the absorbed solar heat. Air is used as a fluidizing medium in this process to enhance particle heat transfer and mixing. The effectiveness of this design lies in its ability to manage solar flux conditions and ensure high solar-to-thermal receiver efficiency. A 100-kW prototype is currently being tested at the King Saud University in Saudi Arabia to assess the receiver performance. A range of modeling analyses for the optical, thermal, and mechanical effects were conducted to assess the performance of the receiver under on-sun conditions. The solar flux resulting from the KSU heliostat field was modeled using NREL SolTrace software and produced up to 600 kW/m2 at the receiver aperture. The solar flux absorbed on the receiver walls was then used within a computational fluid dynamics (CFD) model to predict wall temperature distributions along with radiation and convection loss. A two-phase CFD model was developed for the fluidized bed of silica sand inside the receiver panels to predict local wall-to-particle heat transfer coefficients, particle temperature distributions, and outlet temperature of the particles. We have also conducted analyses to understand the thermomechanical behavior of these innovative enclosed light-trapping solar receivers optimized for particle heating. We used finite element analysis (FEA) to predict the receiver's performance using temperature distributions obtained from CFD and based on the resulting stress profiles, evaluated creep-fatigue damage with a goal of achieving a 30-year service life. Analysis showed a significant impact of the particle-to-wall heat transfer coefficients (HTCs) on receiver performance, with higher HTCs resulting in reduced stress and increased lifespan. For instance, when using Inconel 740H, increasing the HTC from 800 W/m2 K to 1400 W/m2 K increased the creep life from 4,000 hours to over 100,000 hours. This highlights the importance of understanding and optimizing heat transfer in the design of high-efficiency receivers.

14 SOLAR ENERGY↗

Optical control of orbital magnetism in magic-angle twisted bilayer graphene

Flat bands in twisted graphene structures host various strongly correlated and topological phenomena. Optically probing and controlling them can reveal important information such as symmetry and dynamics, but this has been challenging due to the small energy gap compared with optical wavelengths. Here, in this study, we report on the near-infrared optical control of orbital magnetism and associated anomalous Hall effects in a magic-angle twisted bilayer graphene on a monolayer WSe 2 device. We demonstrate control over the hysteresis and amplitude of the anomalous Hall effect near integer moiré fillings using circularly polarized light. By modulating the light helicity, we observe periodic modulation of the transverse resistance in a wide range of fillings, indicating light-induced orbital magnetization through a large inverse Faraday effect. At the transition between metallic and anomalous Hall effect regimes, we also reveal large and random switching of the Hall resistivity, which we attribute to the light-tuned percolating cluster of magnetic domains. Our results demonstrate the potential of the optical manipulation of correlation and topology in moiré structures.

Persky, Eylon [Stanford Univ., CA (United States);↗

First observation of 28 O

Subjecting a physical system to extreme conditions is one of the means often used to obtain a better understanding and deeper insight into its organization and structure. In the case of the atomic nucleus, one such approach is to investigate isotopes that have very different neutron-to-proton ($N/Z$) ratios than in stable nuclei. Light, neutron-rich isotopes exhibit the most asymmetric $N/Z$ ratios and those lying beyond the limits of binding, which undergo spontaneous neutron emission and exist only as very short-lived resonances (about 10 -21 s), provide the most stringent tests of modern nuclear-structure theories. Here we report on the first observation of 28 O and 27 O through their decay into 24 O and four and three neutrons, respectively. The 28 O nucleus is of particular interest as, with the $Z$ = 8 and $N$ = 20 magic numbers, it is expected in the standard shell-model picture of nuclear structure to be one of a relatively small number of so-called ‘doubly magic’ nuclei. Furthermore, both 27 O and 28 O were found to exist as narrow, low-lying resonances and their decay energies are compared here to the results of sophisticated theoretical modelling, including a large-scale shell-model calculation and a newly developed statistical approach. In both cases, the underlying nuclear interactions were derived from effective field theories of quantum chromodynamics. Finally, it is shown that the cross-section for the production of 28 O from a 29 F beam is consistent with it not exhibiting a closed $N$ = 20 shell structure.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Thermal decomposition of neptunyl ammonium nitrate: mechanistic insights and structural characterization of the Np 2 O 5 intermediate phase

Neptunium (Np) possesses a rich and unique chemistry that often diverges from other actinide elements yet remains relatively underexplored compared with the other light actinides. A resurgence of interest in Np has been spurred by the application of 237 Np for plutonium-238 ( 238 Pu) production for use in radioisotope thermoelectric generators (RTGs), necessitating evaluation of Np chemical reactions and materials. The work presented here studied the thermal decomposition of neptunyl ammonium nitrate (NH 4 Np VI O 2 (NO 3 ) 3 ) for synthesis of neptunium dioxide (NpO 2 ), which is the target material used for production of 238 Pu. Additionally, structural characterization of the intermediate solid Np pentoxide (Np 2 O 5 ) was performed. Advanced solid-state characterization techniques, including simultaneous thermal analysis (STA), powder X-ray diffraction (pXRD), Raman spectroscopy, and density functional theory (DFT) modeling have been combined to study the reaction pathways. Analysis revealed that NH 4 Np VI O 2 (NO 3 ) 3 thermally decomposes to a proposed neptunyl nitrate intermediate, followed by Np 2 O 5 and finally NpO 2 , all within the temperature range of 150 °C–600 °C. Further characterization of the pentoxide intermediate provided the first Raman spectra of pure-phase Np 2 O 5 and associated DFT modeling confirmed Raman peak assignments for this phase. These findings provide mechanistic information to advance production of the critical radioisotope 238 Pu and advance the state of knowledge on Np materials chemistry using modern characterization techniques.

Lawson, Kathryn M. [Oak Ridge National Laboratory ↗

Understanding the stability of a plastic‐degrading Rieske iron oxidoreductase system

Abstract Rieske oxygenases (ROs) are a diverse metalloenzyme class with growing potential in bioconversion and synthetic applications. We postulated that ROs are nonetheless underutilized because they are unstable. Terephthalate dioxygenase (TPA DO PDB ID 7Q05 ) is a structurally characterized heterohexameric α 3 β 3 RO that, with its cognate reductase (TPA RED ), catalyzes the first intracellular step of bacterial polyethylene terephthalate plastic bioconversion. Here, we showed that the heterologously expressed TPA DO /TPA RED system exhibits only ~300 total turnovers at its optimal pH and temperature. We investigated the thermal stability of the system and the unfolding pathway of TPA DO through a combination of biochemical and biophysical approaches. The system's activity is thermally limited by a melting temperature ( T m ) of 39.9°C for the monomeric TPA RED , while the independent T m of TPA DO is 50.8°C. Differential scanning calorimetry revealed a two‐step thermal decomposition pathway for TPA DO with T m values of 47.6 and 58.0°C (Δ H = 210 and 509 kcal mol −1 , respectively) for each step. Temperature‐dependent small‐angle x‐ray scattering and dynamic light scattering both detected heat‐induced dissociation of TPA DO subunits at 53.8°C, followed by higher‐temperature loss of tertiary structure that coincided with protein aggregation. The computed enthalpies of dissociation for the monomer interfaces were most congruent with a decomposition pathway initiated by β‐β interface dissociation, a pattern predicted to be widespread in ROs. As a strategy for enhancing TPA DO stability, we propose prioritizing the re‐engineering of the β subunit interfaces, with subsequent targeted improvements of the subunits.

59 BASIC BIOLOGICAL SCIENCES↗

Tantalum pentoxide: a new material platform for high-performance dielectric metasurface optics in the ultraviolet and visible region

Dielectric metasurfaces, composed of planar arrays of subwavelength dielectric structures that collectively mimic the operation of conventional bulk optical elements, have revolutionized the field of optics by their potential in constructing high-efficiency and multi-functional optoelectronic systems on chip. The performance of a dielectric metasurface is largely determined by its constituent material, which is highly desired to have a high refractive index, low optical loss and wide bandgap, and at the same time, be fabrication friendly. Here, we present a new material platform based on tantalum pentoxide (Ta 2 O 5 ) for implementing high-performance dielectric metasurface optics over the ultraviolet and visible spectral region. This wide-bandgap dielectric, exhibiting a high refractive index exceeding 2.1 and negligible extinction coefficient across a broad spectrum, can be easily deposited over large areas with good quality using straightforward physical vapor deposition, and patterned into high-aspect-ratio subwavelength nanostructures through commonly-available fluorine-gas-based reactive ion etching. We implement a series of high-efficiency ultraviolet and visible metasurfaces with representative light-field modulation functionalities including polarization-independent high-numerical-aperture lensing, spin-selective hologram projection, and vivid structural color generation, and the devices exhibit operational efficiencies up to 80%. Our work overcomes limitations faced by scalability of commonly-employed metasurface dielectrics and their operation into the visible and ultraviolet spectral range, and provides a novel route towards realization of high-performance, robust and foundry-manufacturable metasurface optics.

36 MATERIALS SCIENCE↗

Cu(II) Stability and UV-Induced Electron Transfer in a Metal–Organic Hybrid: An EPR, DFT, and Crystallographic Characterization of Copper-Doped Zinc Creatininium Sulfate

Single-crystal X-ray diffraction and electron paramagnetic resonance (EPR) spectroscopic experiments, complemented by quantum chemical DFT calculations, were carried out on the copper-doped metal–organic hybrid and Tutton salt analogue zinc creatininium sulfate to determine its crystal structure, to characterize the electronic structure of the doped Cu(II) binding site, and to propose a pathway for an excited-state, proton-coupled electron transfer (PCET) process in UV-exposed crystals. The crystal structure is isomorphous to that of cadmium creatininium sulfate, which has the transition ion, not in direct coordination with the creatinine, but forming a hexahydrate complex, which is bridged to a creatininium through an intervening sulfate ion. The EPR g (2.446, 2.112, 2.082) and copper hyperfine (A Cu : -327, -59.6, 10.8 MHz) tensor parameters are consistent with doped copper replacing host zinc in the metal–hexahydrate complex. These parameters are similar to those observed for copper hexahydrate in doped Tutton salt systems at low temperature, where the unpaired electron occupies mainly the copper 3d x 2 –y 2 orbital. At room temperature in the Tutton systems, vibration couplings stemming from a dynamic Jahn–Teller effect cause tensor averaging which results in a reduction in their maximum g-tensor and hyperfine tensor values. However, like for the doped isomorphous Cd creatinine crystal, the Cu(II) EPR exhibits little, or no room temperature averaging compared to its low temperature pattern. Samples exposed to 254 nm UV light generate a carbon-centered free radical species, characterized by an isotropic g-tensor (g = 2.0029) and an alpha-proton hyperfine coupling (-24 -14 +4 G). These parameters identify it as a creatinine radical cation formed by the oxidative release of one of its C2 methylene hydrogens. DFT calculations confirm the unpaired electronic structures of both the Cu(II) site and free radical. The growth in radical concentration with an increase in the UV exposure time coincides with a decrease in the copper EPR signal, indicating a coupled light-induced oxidation reduction process. A comparison of the crystal structure with the EPR parameters and DFT results provides evidence for a UV-induced PCET.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Influence of growth temperature of KTiOAsO{sub 4} single crystals on their physicochemical parameters and formation of domain structures

A potassium titanyl arsenate (KTiOAsO{sub 4}, KTA) crystal 50 × 80 × 60 mm in size has been grown by upgraded Czochralski method from flux (TGGS) with a decrease in temperature from 900 to 770 °C during pulling. It is shown that the spectroscopic properties of the parts of KTA crystals grown at 900 and 770 °C are close, whereas the electrical conductivity of the low-temperature (770 °C) KTA part turned out to be an order of magnitude lower than that of the high-temperature part. Visualisation of the domain structure by second-harmonic generation microscopy revealed a more efficient domain intergrowth (throughout the sample) in the low-temperature KTA, which is important for forming a regular domain structure (RDS) in a KTA-based nonlinear optical element. It is established that the quantum efficiency of parametric generation of light in the RDS formed in low-temperature KTA is several times higher than in the case of high-temperature KTA. The results obtained are important for optimising RDS parameters. (paper)

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗